Unless you think Shanahan as a coach changed when somebody flipped a switch on him then there is a relationship. This isn't like talking about players where they can come and go or get a lot older and have physical skills diminish in that time period. How many good new coaches hit it good for a few years and then fizzle in the NFL only to be fired? There are countless number of them, even a few in the NFL right now, even some that have been fired just this year or probably will be fired in a year or two, but coaches that can constantly win and have a good teams most of the time year after year are rare. A lot of teams and there fans would kill to have 8-8 be considered a pretty bad year for them in a 15 year span. Even if you do want him gone who are you going to find to replace him that's better.
As far as playoff victories go it isn't a good indicator of how good a coach is necessarily, especially when the circumstances are taken into account. You can hardly blame Shanahan for fielding good teams in the years where Indianapolis and New England or even San Diego were rolling over everybody or where they had to compete in the AFC. A lot of those years if they were in the NFC they would have been first round byes and probably went to the Superbowl at least once. There was also a time where Indy, New England, and Denver had a rock paper scissors things going on where Indy always seemed to beat Denver, Denver always seemed to beat New England, and New England always seemed to beat Indy. It makes me wonder what would have happened if Denver faced New England some of those times in those playoff years where they faced Indy and got beat.